In this paper we apply neural networks with local converging inputs (NNLCI), originally introduced in [arXiv:2109.09316], to solve the two dimensional Maxwell's equation around perfect electric conductors (PECs). The input to the networks consist of local patches of low cost numerical solutions to the equation computed on two coarse grids, and the output is a more accurate solution at the center of the local patch. We apply the recently developed second order finite difference method [arXiv:2209.00740] to generate the input and training data which captures the scattering of electromagnetic waves off of a PEC at a given terminal time. The advantage of NNLCI is that once trained it offers an efficient alternative to costly high-resolution conventional numerical methods; our numerical experiments indicate the computational complexity saving by a factor of $8^3$ in terms of the number of spatial-temporal grid points. In contrast with existing research work on applying neural networks to directly solve PDEs, our method takes advantage of the local domain of dependence of the Maxwell's equation in the input solution patches, and is therefore simpler, yet still robust. We demonstrate that we can train our neural network on some PECs to predict accurate solutions to different PECs with quite different geometries from any of the training examples.
Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）
The effect of quadrature rules on finite element solutions of Maxwell variational problems. Consistency estimates on meshes with straight and curved elements
Stochastic domain decomposition for the solution of the two-dimensional magnetotelluric problem
Department of Mathematics and Statistics, Memorial University of Newfoundland（纪念大学数学与统计系）；Department of Earth Sciences, Memorial University of Newfoundland（纪念大学地球科学系）
Solving the multi-frequency electromagnetic inverse source problem by the Fourier method
School of Mathematics, Jilin University（吉林大学数学学院）；Department of Mathematics, Harbin Institute of Technology（哈尔滨工业大学数学系）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
Continuous Finite Element Method For Maxwell Eigenvalue Problems With Regular Decomposition Technique
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Two operator splitting methods for three-dimensional stochastic Maxwell equations with multiplicative noise
China University of Mining and Technology（中国矿业大学）；Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；Shanghai Zhangjiang Institute of Mathematics（上海张江数学研究院）
An unconditionally convergent discontinuous Galerkin neural network method activated by plane waves for Helmholtz equation and Maxwell's equations
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
A Crank-Nicolson Finite Element Method and the Optimal Error Estimates for the modified Time-dependent Maxwell-Schrödinger Equations
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）；LSEC, NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所）
A multi-moment scheme for the two dimensional Maxwell's equations
Center for Research in Scientific Computation, North Carolina State University（北卡罗来纳州立大学科学计算研究中心）；Department of Mathematics, North Carolina State University（北卡罗来纳州立大学数学系）
